Infinity is contained within the concept of options which are some of the mechanisms that serve as the foundation of Existence and Non Existence. We experience this through Free Will.
18:11 "Between any two real numbers there's a rational number." You guys are killing me! How do I even begin to get that into perspective with the fact that there are more reals than rationals? Between every Real/post, there is a rational/fence, but the difference in total posts and total fences is uncountable.
@@Grateful92 "Most of what we are is non physical, though, our lowest form is physical. All life on our planet has the lowest form, the Body. Our Body is an Animal and the other type of Body on our planet is a Plant. Bodies are bound absolutely to Natural waL (spelt backwards) which is the lowest form of true Law. Natural waL (spelt backwards) is a localised form of Law and is derived from the Laws of Nature. Natural waL (spelt backwards) is the finite and specific foundational control structure ordering the actions and interactions of species, members of species, and the material sources of a planet. The lowest non physical form of what we are is the Mind, which is a Process. There are other forms of life on our planet that have both a Body and a Mind, however, so far as we currently know, there are no Plants and only some Animals that have a Body and a Mind. The lowest forms of Mind, Instinct and Emotion, are predominantly bound to Natural Law. The next higher form of Mind is Intellect which is bound predominantly to the Laws of Nature. Intuition, the highest form of Mind, can be bound or not to both Natural Law and the Laws of Nature separately or together, or to higher forms of Law altogether. Intuition is the truest guide for our Selves. The next non physical form of what we are is the Self, which is an Awareness. There are relatively few other forms of life on our planet that have a Self. The Self is not bound to any form of Law other than One's Own Law. It is the only form of Law that cannot be violated. The foundation of what we are is the highest non physical form of what we are. The highest form of what we are is the Being, which is an Existence. The Being is not bound to any form of Law originating within Existence. The Being is bound absolutely to The Law. Existence, and the Laws of Nature which are the finite and specific foundational control structure ordering the actions and interactions of all elements within Existence, cannot Be without The Law being The Law. So, what is The Law? In a word, The Law is options. Definition option: a thing that is or may be chosen. The word 'option' does convey the idea of The Law in its most basic sense but does not clarify all of what The Law is. Free Will does describe how our species experiences The Law but does not convey all of what The Law is. In clarifying what The Law is; The capitalised form of the word 'The' indicates the following noun is a specific thing. Law is the finite and specific foundational control structure ordering the actions and interactions of all elements subordinate. Together, the words 'The' and 'Law' (in that exact order,) is a proper noun indicating; the singular form of Law that all other forms of Law and all other Laws are founded upon, the singular foundation upon which Existence is founded, the singular foundation upon which Non Existence is founded, the singular foundation connecting Existence to Non Existence, the concept of options, and Free Will. However one thinks, believes, guesses, hopes, or "knows", whether by a gnaBgiB (spelt backwards), a creation story, a computer program, an expansion of consciousness, or whatever means by which Existence could have come to Be, the option for Existence to not Be also exists. Existence and Non Existence, the original options connected by the very concept of options, connected by The Law. Outside of space and before time. Extra-Existential. As we experience The Law in our Being, The Law is Free Will. The First Protector of The Law is Freely Given Consent. The First Violation of The Law is Theft of Consent." - Goho-tekina Otoko
what a quote. Honestly there is something about the mathematical language that doesn't fit for every different brain and mental wiring out there. In school it was the hardest thing with all teachers except one who had a different way of explaining things that just flowed. The key really is that we're using a poor method of description a poor method of interpretation of the world
That strikes me as fleshing out to be: 1. To understand would require truth. But absolute truth is not accessible to us. As such, there exists no definitive handle upon which to mathematically anchor one's self. [In alignment with Godel's incompleteness theorems] 2. Notwithstanding, we must get on with it. Getting on with it comes via routine. Use routine as an anchor. We have a plethora of routines to choose from. Any anxiety about not having a handle to grab ahold of will surely give way to one's commitment to routine. Very much in line with Mermin's Shut up and Calculate quip.
Brady is low-key one of the best interviewers and students ever. I always get the feeling that he is way more knowledgeable than he lets on just by the quality of his questions and the way he steers the conversations.
absolutely, and having watched this channel for more than half a decade now, you can actually notice him getting more and more knowledgeable in all fields of math, just like us watching along
On Brady’s “why 2” question. Yes it doesn’t matter numerically, but it is not arbitrary. It represents the cardinality of a power set - the set of all subsets of a set. To form a subset of a set X you need to make a binary choice (in/out) for each element of X. So 2^X is a common notation for the power set, and then |2^X|=2^|X|
EXACTLY. Thank you. I don't know why power sets weren't mentioned anywhere here when they are key to understanding these concepts on a more than "I just said so" level.
Yes and just to really spell it out, that notation is expressing the fact that the real numbers have the same cardinality as the power set of the natural numbers
Missed opportunity to tell Brady that his improvised term "graspable" has a formal equivalent, which is "countable". The natural and rational numbers are countably infinite; the real numbers are not.
@@michaelsmith4904no you can't, far from it. even after counting off 1 real number every nanosecond forever you'd have counted aleph 0 numbers, while there are aleph 1 ahead
@@michaelsmith4904not really, as even if you counted them all, you'd still be able to make an entirely new unique real number to add to it. So you can always add another number to the set so you can never have an entire set to count, hence the uncountable.
If your model of set theory has an inaccessible cardinal, you can define the universe of sets up to that cardinality (in the von Neumann hierarchy). That universe doesn't contain its own cardinality, and there is no set in the universe as large as the universe itself, so you do essentially "run out of sets." Or if you use the whole universe V, you can discuss in a philosophical sense the "size" of V, and that can't possibly be the size of a set (because that would have to be a universal set). Rather, it's the size of a proper class. It's consistent that all proper classes have the same size, but it's also consistent that they have different sizes. But even if they all have the same size, that size is not a cardinal, because you can't form an equivalence class of proper classes.
@@EebstertheGreat If your model of set theory has an inaccessible cardinal, the universe contains much more then just cardinals up to that inaccessible cardinal.
Just want to quickly mention that the "2 to the Aleph_0" IMO comes from taking power sets. Take a (finite) set X, and then consider the set of all subsets of X. This new set is called the power set and has precisely 2^|X| elements, with |X| denoting the number of elements of X. And this will always be strictly larger than the original set; even when considering infinite once. Hence the 2 to the power of...part :)
@@galoomba5559Wait isn't it the other way around? Shouldn't it be the functions of X to {0,1}? In that case the isomorphism is very simple. Given a subset Y, a function f_Y and an input x, return 0 if x is not in Y and 1 if x is in Y. Now clearly the functions from X to {0,1} and the subsets of X are 1-to-1.
Yes, all of this... this video seemed lazy and inaccessible to people... not up to the usually quality of Numberphile. There are so many ways to think if the class of infinite cardinalities and how to show that the cardinals do indeed get larger, which was just kind of presented axiomatically here without any constructive proof.
The example I would give as a response to the question whether this is used: Turing showed that the size of the computable numbers is aleph 0. This immediately implies that non-computable real numbers exist for example the diagonal numbers of the computable numbers. And if you look for the reason you can not compute these you discover the Halting problem.
Although (I believe) there are only countably many numbers that are uncomputable for that specific reason. Since there are only countably many "definable" numbers at all, and the remaining ones are both uncomputable and undefinable.
@@MrCheeze Take the reals and remove any set size aleph-0. Say, the rationals. You are still left with an uncountable amount of numbers, since Cantor's diagonal argument still works with a sequence of irrational numbers, like (π, 2π, 3π, ...) And we can actually define an uncomputable number The sum of 1/TREE(n) or the sum of 1/BB(n) are easy examples of definable but uncomputable numbers By proving there is a bijection between the naturals and a set S of uncomputable numbers, and by defining at least 1 uncomputable number ∉ S, we show that there is a uncountable amount of uncomputable numbers Let S = { [sum(1/BB(n))]^1, [sum(1/BB(n))]^2, [sum(1/BB(n))]^3, ... } Enumerate the elements of S by using the naturals Remember the sum of 1/TREE(n), well this element ∉ S and we already used all the naturals to enumerate S, so there is an uncountable amount of uncomputable numbers ;)
@@thewhitefalcon8539 Yeah, I know. But you didn't mean to allow for the possibility that there could be finitely many uncomputable numbers? Actually, reading your comment again: "countably many uncomputable numbers"? That cannot be right. The computable ones are countably infinite, so the remaining uncomputable ones must be uncountably many in number.
Can you proof this? Even if numberphile has a very positive effect on all living humans and on all humans that will ever live till the end of the universe this number will be quite small compared to infinity...
The question is super interesting, but the answer is somewhat misleading. While it is true that 2^aleph = 3^aleph = aleph^aleph, 2 isn't a random choice. It represents the power set, which is the set of every subset of aleph
True comprehension of infinity is beyond us... but attempting to turn one's own brain into a black hole is always a worthy pursuit, and comprehending infinity is the most fun way to do that, in my opinion.
@@antoniocortijo-rodgers75 For some reason, there is a _widespread_ misconception about the definition of Aleph numbers. Tian is correct. And I assume you're one of the people who have heard someone define Aleph numbers improperly. So I suggest you look up "aleph numbers" and "beth numbers", and in particular, their relation to the Continuum Hypothesis.
I also would be surprised, if he had forgotten the explanation of the other professors. I remember Dr. Grimes calling the Aleph_0 size sets „listable numbers“. Hence you can list natural, whole and rational numbers, they are all the same size. At real numbers you don‘t even know the next number in the list after 0.
@@SmashXano What do you mean by "not knowing the next number in the list"? You can pick any number to be the next number in the list. The point is that any list you make in this way will not contain all the real numbers.
@@galoomba5559 I could be wrong, but it sounds like another way of stating that, no matter which 2 real numbers you choose, you will always be able to find a real number that lies between them in value.
I feel like your real talent with these videos is the questions you ask to prod and pull apart these experts that you interview. You’ve clearly learned a lot over the years and know exactly how to get the most out of your guests. Thanks for all of your hard work Brady!
I love the "Why?" from Asaf when getting Brady to place the rationals against the naturals. It seems so inquisitive and I love this channel for having these conversations as a proxy for us asking the same questions. I hope Brady understands how important these channels are!
17:30 a fun fact I'll never not keep repeating. The rationals have the same size as the natural numbers. Because of the way you measure sizes when you're playing with infinite sets and measures, this means that they have size ZERO in the set of reals. BUT, they are also dense in the reals, meaning you can find a rational number arbitrarily close to any real number. So they're nowhere but also everywhere at the same time in the set of real numbers. 😂🤯
Wouldn't it be more fun to mention how the algebraic numbers are also countable? Not just every rational number, but every single solution to any polynomial with integer coefficients. Every strange thing you can make with addiction, multiplication and integer roots. They're countable.
@@xinpingdonohoe3978 the algebraic numbers - the solutions to any polynomial of any degree - are a cool set as well, for sure. I believe that the computable numbers, which includes e, pi etc, are also the same size. In fact I think there's a Matt Parker video about it on this channel from a few years ago.
'Lazy 8' is when the symbol is used for cattle branding. The mathematical symbol for infinity is called a lemniscate (Latin for 'decorated with ribbons').
@@PilpelAvital 'Losing one glove is certainly painful, but nothing compared to the pain, of losing one, throwing away the other, and finding the first one again.' My favourite Piet Hein quote 😁
After being stuck on some finite math, it was an absolute joy to come home to a Numberphile video on absolute infinity - featuring one of my absolute favorite former office mates :)
Brady, it has to be said you asked great questions in this video. Not just as our voice as the viewer, but great questions as an interviewer of an expert.
I think we can say that there's something special about 2^Aleph_0. Aleph_0 is the cardinality of the natural numbers N,. "2^N" represents the space of functions from N to a two-element set. W.L.O.G. we can take that set to be the set {TRUE, FALSE} - which helps us to see that each way to map N to {TRUE, FALSE} (i.e. each possible such function) defines a subset of the natural numbers - namely the set for which the function returns TRUE. The totality of all possible ways to map N to {TRUE, FALSE} then defines all possible subsets of the natural numbers. This means that the cardinality of this function space *is* the cardinality of the powerset (set of all subsets) of N, P(N). Since Aleph_0 is the cardinality of the naturals, the cardinality of their powerset is exactly 2^Aleph_0 As we have learned in the video, this is the cardinality of the reals - so the cardinality of the reals is exactly the cardinality of the powerset of the naturals.
I really appreciate Brady's ability to ask questions that a) I also find fascinating and b) the interviewees really appreciate and can build on. A great skill.
My goodness, I've fallen in love with that mathematician! He was/is so appealing in his intelligence and his amazing ability to describe complex thoughts and theories. Wow. Very appealing man. lol
I think this could become a series, maybe even including several presenters, who talk about the topics of infinities, set theory, axiom of choice, continuum hypothesis... As a physicist, it's the kind of maths I've not been exposed to but would be really interested in learning more about!
An infinity video about set theory where countability is not once mentioned. Truly an amazing feat. Every time they talked about Aleph not, I expected them to mention that it is a countable infinity.
I think the surreal numbers are really useful for wrapping your head around these different kinds of infinity. IIRC, the sort of sense in which all of the alephs get treated like they're just a different set of natural numbers is a key part of that: you've got all the real numbers on their line, then for each real number there are as many infinitesimal numbers, that are each closer to that real number than any other real number, as there are real numbers; and so on for each of those infinitesimals, etc, all the way down forever; but also in the other direction, all of those original real numbers are closer to a given transfinite number than any other transfinite number is, and there are as many of that class of transfinite numbers as there are reals, each with a whole "real number line" of its own that are closer to that one than to any other; and so on for all of those transfinite numbers, they're all closer to some even greater transfinite number than any other number of that higher class is, etc, all the way up forever. I think it's provable that the surreal numbers are *the* most complete number line there could possibly be: any kind of number no matter how big or small anyone might ever come up with, it's already in the surreals. But then what about numbers that aren't on lines? Complex numbers, and hypercomplex numbers like quaternions and octonions. Those can be "sur" as well, not just the reals! There are surcomplex numbers too, and surhypercomplex numbers like surquaternions and suroctonions. I would love to see a video with someone quickly going over the construction from the empty set all the way up to the suroctonions.
It's not true that the surreals contain every number, in part because "number" is not a well-defined term. The complex numbers aren't contained in the surreals, for example. What is contained in the surreals is every ordered field.
@@galoomba5559 Thank you for clarifying that. I did mean to exclude (hyper)complex numbers from the surreals myself, when I said specifically "number *line*", and then went on to talk about "numbers that aren't on lines".
If number line means an ordered field, the surreals are the most complete number line in that for any other (set-sized) number line, you can find a subset of the surreals that is isomorphic to it. The well-ordered transfinite hierarchy of the ordinals (and cardinals) has to come before the surreal numbers (which are non-well-ordered) can be defined though, because the surreals are defined by induction along the ordinals, and the induction requires well-ordering.
In some systems used to study combinatorial game theory, there are numbers (or nimbers) which are larger in magnitude than any surreal, and numbers which are smaller in magnitude than any surreal, and numbers which are not comparable to (some) surreals. You might check out "Winning Ways for your Mathematical Plays," which, like Knuth's book on the surreals, is kind of fun to read even if you don't care too much about the math.
Brady’s questions almost always leave me impressed with both how sharp he is and how good he is at leading the expert on to explain things in a manner more comprehensible to the viewers. He’s sneakily very, very smart.
17:45 - nice job brady. you didn't tell him you already knew. and that actually is a nice practice, try to think of why those things should feel more natural. then you take a notice of how things change because of that, and with that you can use other things to identify mistakes or problems with your line of thought.
6:02 What a great question. I can't believe I never thought of it myself, but I'm so glad Brady did. Greatest mathematics journalist of all time, for all history.
The fact that you can have Aleph_0 and Aleph_1, both different magnitudes of Infinity, the set of Alephs is countable, and there is an uncountable Aleph, you've now created a set of Alephs. Absolute Infinity is the Proper Class of these objects which is uncountable. It seems like that is just a concept which nullifies any further expansion of Aleph_Omega Sets because we don't have an abstraction which requires any distinction. There are no properties of that Proper Class which make it unique to another Proper Class of Absolute Infinity, so therefore they are equivalent. What I'm unclear about is why we'd need the distinction between Aleph_Omega and Aleph_Omega+1. Is it just because we've decided that the set of Aleph_Omega is countable because it is defined in terms of Natural numbers and therefor countable? Is the notion of Absolute Infinity thereby an artificial construct of our definition, or is there a necessary reason for us to have this distinction? It seems like we could have simply recognized that Aleph_Omega and Aleph_Omega+1 are members of an uncountable set for which we notate using an set of natural numbers which we define as uncountable. Otherwise what is to prevent Aleph_(Omega+Aleph_(Omega+...))?
My thoughts exactly. A lot of "maths" involving infinity becomes absurd and/or quite subjective, I always think twice about the conclusions they postulate
You need to construct sets from other sets. This is the limitation sets to prevent Russell's paradox. Higher infinities some of them cannot be constructed from below so they are no longer sets but they still have describable properties so they are Classes.
@@c1arkj You are saying that nothing is something, but the concept of nothing is not. That's saying that 0 = 1, 1 = 0, 1 ≠ 1, 0 ≠ 0. Do you see the problem with that?
this was a gem of a video. Asaf explaining things clearly (as clearly as he can while keeping it understandable for us!), Brady asking exactly the sort of questions that were needed...
There are mathematicians who are finitists, who insist that any math done with infinity is not legitimate. And there are even ultrafinitists who insist that very large finite numbers (far bigger than anything that would come up in a physical context) are not "real" or legitimate in some sense. It's a minority position though.
The empty set is empty, but the set of the empty set is not, it contains the empty set. Put another way: There is nothing in the empty set, but the set itself is not nothing. (And from that, everything else follows.)
First time I read about omega, omega+1, etc. until finally epsilon, was in Hofstadters _Gödel, Escher, Bach_ . At the time I wasn't even sure if this wasn't just another wordplay between Achilles and the Tortoise. Glad that's sorted out. 😊
Studied law and chemistry. Listening to this makes me amaze how people can engage generations of scientists with word plays that lets them explore numbers again as if they were new.
I always love how Brady asks the exact questions I'm thinking in my head. I never feel frustrated by a numberphile video, because it's almost like I'm conversing directly with the mathematician. It's uncanny how good he is at asking the questions we're all thinking.
@@AlanShi-wm6tw my username? yeah, a friend lent me a book about Georg Cantor years and years ago, and i became fascinated by his work and the concept of there being more than one infinity, and eventually just started using either alephnull or alefnull as my username. even got the aleph null symbol tattooed on my wrist lol
So rare I actually feel a little smarter after watching this video.. This is the very best most understandable declarative explanation of set theory I have ever seen because you can almost touch the empirical with this explanation.
About the "why two?" question around 3:46, there's actually a pretty natural intuition for it. If you try to create a subset out of a set A, you can do so with the application f: x -> { 0 if x not in subset, 1 otherwise That is, one of two possibilities for each item. That is, 2^|A| possible subsets. So the powerset of A has 2^|A| element in it, and for N to R numbers we can prove the relation between their size is that of getting powerset, so it "makes sense" to use the notation. Not sure how well this continues holding for higher infinities but I would expect it to sortof continue making sense. 3^ might also work but wouldn't be minimal.
4:17 They use 2 because of sets. It's the same as the power set just with an exact equivalence. There are two options: in the set vs. not in the set. Exponentiation in set theory: x^y is the set of functions from y to x. This is why in set theory and combinatorics 0^0=1
I was always fascinated by the fact that there are more real numbers between 0 and 1 then there are counting numbers, which is like the simplest example for differently sized infinities
That was a big mind blower moment in undergraduate mathematics for me. First they showed us Cantor's diagonal argument to prove the Rational numbers were strictly smaller than the Reals. Then they proved that between any two real numbers there is a rational number. Completely counterintuitive and shows how intuition breaks down significantly when talking about infinity.
I have caught a glimpse of the shadow of infinity three times in my life, thus far. It is an overwhelming experience which one does not quickly forget.
My brain started hurting when he said that 2 to the power of aleph nought was the same as 3 to the power of aleph nought, and that there are as many Real numbers as there are Complex numbers. When I slice the sphere of Netscape Navigator, I'm only limited by my imagination as to how many "types" of shapes I see: whether it is a ceramic pot, a dumbbells or an igloo.
The towers of hanoi analog was quite crucial to comprehending the whole thing. Those symbols and names I will never be able to intuitively remember or name at a later date, nor will I comprehend everything.
I liked the way this ended. I've thought for a long time that part of the reason analysis is at first so difficult and frustrating is because it's almost impossible to wrap one's head around the fact that the natural numbers are dense in the reals but come from an infinitely smaller set.
More videos and Numberphile podcast featuring Asaf - ua-cam.com/play/PLt5AfwLFPxWJyt0zdvzvDoeL_8pqO0S7p.html
Отметься кто смотрить ролик до конца 🐼......
Infinity is contained within the concept of options which are some of the mechanisms that serve as the foundation of Existence and Non Existence. We experience this through Free Will.
18:11 "Between any two real numbers there's a rational number."
You guys are killing me! How do I even begin to get that into perspective with the fact that there are more reals than rationals? Between every Real/post, there is a rational/fence, but the difference in total posts and total fences is uncountable.
@@starc. Where can I learn more about it!?
@@Grateful92 "Most of what we are is non physical, though, our lowest form is physical. All life on our planet has the lowest form, the Body. Our Body is an Animal and the other type of Body on our planet is a Plant. Bodies are bound absolutely to Natural waL (spelt backwards) which is the lowest form of true Law. Natural waL (spelt backwards) is a localised form of Law and is derived from the Laws of Nature. Natural waL (spelt backwards) is the finite and specific foundational control structure ordering the actions and interactions of species, members of species, and the material sources of a planet.
The lowest non physical form of what we are is the Mind, which is a Process. There are other forms of life on our planet that have both a Body and a Mind, however, so far as we currently know, there are no Plants and only some Animals that have a Body and a Mind. The lowest forms of Mind, Instinct and Emotion, are predominantly bound to Natural Law. The next higher form of Mind is Intellect which is bound predominantly to the Laws of Nature. Intuition, the highest form of Mind, can be bound or not to both Natural Law and the Laws of Nature separately or together, or to higher forms of Law altogether. Intuition is the truest guide for our Selves.
The next non physical form of what we are is the Self, which is an Awareness. There are relatively few other forms of life on our planet that have a Self. The Self is not bound to any form of Law other than One's Own Law. It is the only form of Law that cannot be violated.
The foundation of what we are is the highest non physical form of what we are. The highest form of what we are is the Being, which is an Existence. The Being is not bound to any form of Law originating within Existence. The Being is bound absolutely to The Law.
Existence, and the Laws of Nature which are the finite and specific foundational control structure ordering the actions and interactions of all elements within Existence, cannot Be without The Law being The Law.
So, what is The Law?
In a word, The Law is options.
Definition
option: a thing that is or may be chosen.
The word 'option' does convey the idea of The Law in its most basic sense but does not clarify all of what The Law is.
Free Will does describe how our species experiences The Law but does not convey all of what The Law is.
In clarifying what The Law is;
The capitalised form of the word 'The' indicates the following noun is a specific thing.
Law is the finite and specific foundational control structure ordering the actions and interactions of all elements subordinate.
Together, the words 'The' and 'Law' (in that exact order,) is a proper noun indicating;
the singular form of Law that all other forms of Law and all other Laws are founded upon,
the singular foundation upon which Existence is founded,
the singular foundation upon which Non Existence is founded,
the singular foundation connecting Existence to Non Existence,
the concept of options, and
Free Will.
However one thinks, believes, guesses, hopes, or "knows", whether by a gnaBgiB (spelt backwards), a creation story, a computer program, an expansion of consciousness, or whatever means by which Existence could have come to Be, the option for Existence to not Be also exists. Existence and Non Existence, the original options connected by the very concept of options, connected by The Law. Outside of space and before time. Extra-Existential.
As we experience The Law in our Being,
The Law is Free Will.
The First Protector of The Law is Freely Given Consent.
The First Violation of The Law is Theft of Consent."
- Goho-tekina Otoko
"In mathematics, you don't understand things, you just get used to them." - John von Neuman
I never heard this quote before, but I love it!
I never did understand that quote, but I eventually got used to it.
what a quote. Honestly there is something about the mathematical language that doesn't fit for every different brain and mental wiring out there. In school it was the hardest thing with all teachers except one who had a different way of explaining things that just flowed. The key really is that we're using a poor method of description a poor method of interpretation of the world
"There's a trick you can use in mathematics called not worrying about it." - Matt Parker
@@SellymeYT Sounds more like an Andrew Ng quote.
That strikes me as fleshing out to be:
1. To understand would require truth. But absolute truth is not accessible to us. As such, there exists no definitive handle upon which to mathematically anchor one's self. [In alignment with Godel's incompleteness theorems]
2. Notwithstanding, we must get on with it. Getting on with it comes via routine. Use routine as an anchor. We have a plethora of routines to choose from. Any anxiety about not having a handle to grab ahold of will surely give way to one's commitment to routine.
Very much in line with Mermin's Shut up and Calculate quip.
Brady is low-key one of the best interviewers and students ever. I always get the feeling that he is way more knowledgeable than he lets on just by the quality of his questions and the way he steers the conversations.
Absolutely had the same feeling for years now 😄
pssst, hey buddy.... wanna buy a Numberphile script? ;-)
fundamental attribution error (that's a real thing, google it)
absolutely, and having watched this channel for more than half a decade now, you can actually notice him getting more and more knowledgeable in all fields of math, just like us watching along
Yes. He'd make a great news reporter.
I love that Numberphile combines both modern quality of presentation and old school vibe of filming which is quite comforting in a way.
yes
The best of both worlds.
It's a very specific era of old school, the handheld 'camcorder' style is kind of mid-90s to 2000's era old school
this is a numberphile signature and should never be changed
Even the animations remind me of the CGI visualizations of 90s/00s math educational films (like 'Outside In') in the best way
"Just infinity. You say it like it's just a trivial thing"
"YES."
It's like the gigachad meme
"...you know, it's an everyday thing one encounters in their life. Nothing too crazy."
Infinity is an occult symbol.
This is SET THEORY. Infinity is super normal around here.
On Brady’s “why 2” question. Yes it doesn’t matter numerically, but it is not arbitrary. It represents the cardinality of a power set - the set of all subsets of a set. To form a subset of a set X you need to make a binary choice (in/out) for each element of X. So 2^X is a common notation for the power set, and then |2^X|=2^|X|
EXACTLY. Thank you. I don't know why power sets weren't mentioned anywhere here when they are key to understanding these concepts on a more than "I just said so" level.
Yes and just to really spell it out, that notation is expressing the fact that the real numbers have the same cardinality as the power set of the natural numbers
Missed opportunity to tell Brady that his improvised term "graspable" has a formal equivalent, which is "countable". The natural and rational numbers are countably infinite; the real numbers are not.
I wish Dr Karagila explicitly mentioned that. It would have been a perfect conclusion. Still a great video
the weird thing is even though you can't count the real numbers, you can come arbitrarily close...
@@michaelsmith4904no you can't, far from it. even after counting off 1 real number every nanosecond forever you'd have counted aleph 0 numbers, while there are aleph 1 ahead
@@michaelsmith4904 what do you mean?
@@michaelsmith4904not really, as even if you counted them all, you'd still be able to make an entirely new unique real number to add to it. So you can always add another number to the set so you can never have an entire set to count, hence the uncountable.
15:13 "Now I'm asking you, a set theorist, who deals with infinity every day, and throws around infinity like pieces of candy..." Legendary
9:53
- It gets bigger and bigger until eventually you "run out of sets".
- How can you ran out?
- Exactly!
Hilarious!
Yeah, that was like something out of Catch-22.
If your model of set theory has an inaccessible cardinal, you can define the universe of sets up to that cardinality (in the von Neumann hierarchy). That universe doesn't contain its own cardinality, and there is no set in the universe as large as the universe itself, so you do essentially "run out of sets." Or if you use the whole universe V, you can discuss in a philosophical sense the "size" of V, and that can't possibly be the size of a set (because that would have to be a universal set). Rather, it's the size of a proper class. It's consistent that all proper classes have the same size, but it's also consistent that they have different sizes. But even if they all have the same size, that size is not a cardinal, because you can't form an equivalence class of proper classes.
Classic Numberphile moment for sure
@@EebstertheGreat If your model of set theory has an inaccessible cardinal, the universe contains much more then just cardinals up to that inaccessible cardinal.
@@TianYuanEX If κ is an inaccessible cardinal, then V_κ is a model of ZFC. It contains everything in the cumulative hierarchy before κ.
Just want to quickly mention that the "2 to the Aleph_0" IMO comes from taking power sets. Take a (finite) set X, and then consider the set of all subsets of X. This new set is called the power set and has precisely 2^|X| elements, with |X| denoting the number of elements of X. And this will always be strictly larger than the original set; even when considering infinite once. Hence the 2 to the power of...part :)
And the reason why it's notated that way is because the power set is isomorphic to the set of all functions from the set to a set with two elements.
@@galoomba5559True, I forgot to mention that!
@@galoomba5559Wait isn't it the other way around? Shouldn't it be the functions of X to {0,1}? In that case the isomorphism is very simple. Given a subset Y, a function f_Y and an input x, return 0 if x is not in Y and 1 if x is in Y. Now clearly the functions from X to {0,1} and the subsets of X are 1-to-1.
@@SolMasterzzz Of course, my bad
Yes, all of this... this video seemed lazy and inaccessible to people... not up to the usually quality of Numberphile.
There are so many ways to think if the class of infinite cardinalities and how to show that the cardinals do indeed get larger, which was just kind of presented axiomatically here without any constructive proof.
The example I would give as a response to the question whether this is used: Turing showed that the size of the computable numbers is aleph 0. This immediately implies that non-computable real numbers exist for example the diagonal numbers of the computable numbers. And if you look for the reason you can not compute these you discover the Halting problem.
Although (I believe) there are only countably many numbers that are uncomputable for that specific reason. Since there are only countably many "definable" numbers at all, and the remaining ones are both uncomputable and undefinable.
@@MrCheeze Take the reals and remove any set size aleph-0. Say, the rationals.
You are still left with an uncountable amount of numbers, since Cantor's diagonal argument still works with a sequence of irrational numbers, like (π, 2π, 3π, ...)
And we can actually define an uncomputable number
The sum of 1/TREE(n) or the sum of 1/BB(n) are easy examples of definable but uncomputable numbers
By proving there is a bijection between the naturals and a set S of uncomputable numbers, and by defining at least 1 uncomputable number ∉ S, we show that there is a uncountable amount of uncomputable numbers
Let S = { [sum(1/BB(n))]^1, [sum(1/BB(n))]^2, [sum(1/BB(n))]^3, ... }
Enumerate the elements of S by using the naturals
Remember the sum of 1/TREE(n), well this element ∉ S and we already used all the naturals to enumerate S, so there is an uncountable amount of uncomputable numbers ;)
@@MrCheeze Just for clarity, when you write "countable", you presumably mean "countably infinite"?
@@landsgevaer countable cardinalities are finite natural numbers and aleph-null
@@thewhitefalcon8539 Yeah, I know. But you didn't mean to allow for the possibility that there could be finitely many uncomputable numbers?
Actually, reading your comment again: "countably many uncomputable numbers"? That cannot be right. The computable ones are countably infinite, so the remaining uncomputable ones must be uncountably many in number.
The quality of numberphile = absolute infinity
Nuh uh
Can you proof this?
Even if numberphile has a very positive effect on all living humans and on all humans that will ever live till the end of the universe this number will be quite small compared to infinity...
@@red.aries1444 my source is dude trust me bro
@@red.aries1444*resurrects Ernest Zermello*
There are no upper bounds on opinions 😋
3:56
Brady is always so good at asking the most interesting questions... I'd never think to question that but I'm so glad he did!
The question is super interesting, but the answer is somewhat misleading. While it is true that 2^aleph = 3^aleph = aleph^aleph, 2 isn't a random choice. It represents the power set, which is the set of every subset of aleph
@@shmendusel I'm surprised he didn't remark this!
asaf is such a wonderful presenter, i feel like he could answer any question brady throws at him!
True comprehension of infinity is beyond us... but attempting to turn one's own brain into a black hole is always a worthy pursuit, and comprehending infinity is the most fun way to do that, in my opinion.
2:58 : “not to scale ... obviously” : haha
Not to scale, naturally.
The least controversial statement in the video at 4:27
> "There is nothing between aleph null and aleph one."
That was my thought too… isn’t he just straight up assuming the continuum hypothesis there?
@@WaffleAbuser No, he isn't, you are thinking about 2^(aleph_0). Aleph_1 is literally defined as the smallest uncountable infinity.
@@TianYuanEXyou’re just wrong tho lol
@@antoniocortijo-rodgers75 For some reason, there is a _widespread_ misconception about the definition of Aleph numbers. Tian is correct. And I assume you're one of the people who have heard someone define Aleph numbers improperly. So I suggest you look up "aleph numbers" and "beth numbers", and in particular, their relation to the Continuum Hypothesis.
@@antoniocortijo-rodgers75 What am I wrong about?
hope we get another session with Asaf about the axiom of choice!
17:30 Brady's so eloquent, but we all know he's known the answer for quite some time :D
I also would be surprised, if he had forgotten the explanation of the other professors. I remember Dr. Grimes calling the Aleph_0 size sets „listable numbers“. Hence you can list natural, whole and rational numbers, they are all the same size. At real numbers you don‘t even know the next number in the list after 0.
his answer is incredibly honest, whether he remembers all the videos he did on this subject or not... he really is building an intuition for it!
@@SmashXano What do you mean by "not knowing the next number in the list"? You can pick any number to be the next number in the list. The point is that any list you make in this way will not contain all the real numbers.
@@galoomba5559 I could be wrong, but it sounds like another way of stating that, no matter which 2 real numbers you choose, you will always be able to find a real number that lies between them in value.
@@yudasgoat2000 That's also true for the rational numbers, and they are countable.
I feel like your real talent with these videos is the questions you ask to prod and pull apart these experts that you interview. You’ve clearly learned a lot over the years and know exactly how to get the most out of your guests. Thanks for all of your hard work Brady!
I love the "Why?" from Asaf when getting Brady to place the rationals against the naturals. It seems so inquisitive and I love this channel for having these conversations as a proxy for us asking the same questions.
I hope Brady understands how important these channels are!
17:30 a fun fact I'll never not keep repeating. The rationals have the same size as the natural numbers. Because of the way you measure sizes when you're playing with infinite sets and measures, this means that they have size ZERO in the set of reals. BUT, they are also dense in the reals, meaning you can find a rational number arbitrarily close to any real number. So they're nowhere but also everywhere at the same time in the set of real numbers. 😂🤯
Wouldn't it be more fun to mention how the algebraic numbers are also countable? Not just every rational number, but every single solution to any polynomial with integer coefficients. Every strange thing you can make with addiction, multiplication and integer roots. They're countable.
@@xinpingdonohoe3978 Huh, that _is_ fun and surprising! Thanks!
@@xinpingdonohoe3978 the algebraic numbers - the solutions to any polynomial of any degree - are a cool set as well, for sure. I believe that the computable numbers, which includes e, pi etc, are also the same size. In fact I think there's a Matt Parker video about it on this channel from a few years ago.
It's as if there was an infinitely thin silk textile that can let things go through and yet block everything
Between every two irrational numbers there is a rational number and between every two rational numbers there is an irrational number.
Yeah, well, whatever the thumbnail is, +1. I win
Did you just timeshift infinity?
Wow, so creative. You're definitely the first person to think of that. It definitely applies to infinite numbers.
@@overestimatedforesightI'm sure it was just intended as a silly joke. :)
oh yeah, well, your number +0.1. i win.
8:12
I like this guy's style and topic, throwing nice trivia like "lazy eight" and that last quote around among profound math.
I've never heard that called a "lazy eight" before...
but I kinda love it
“Lazy 8” actually comes from branding livestock. Which is, if you squint, a kind of heraldry.
'Lazy 8' is when the symbol is used for cattle branding. The mathematical symbol for infinity is called a lemniscate (Latin for 'decorated with ribbons').
Infinity isn't 8 on the side. 8 is infinity standing on end! - Piet Hein.
@@PilpelAvital 'Losing one glove is certainly painful, but nothing compared to the pain, of losing one, throwing away the other, and finding the first one again.'
My favourite Piet Hein quote 😁
Animations above and beyond, Brady. 3Blue1Brown will be looking to his laurels!
Pete McPartlan did the animations. 👍🏻
This was a very insightful video not just about infinity but also why it is important to have such advanced level of maths
After being stuck on some finite math, it was an absolute joy to come home to a Numberphile video on absolute infinity - featuring one of my absolute favorite former office mates :)
Brady, it has to be said you asked great questions in this video. Not just as our voice as the viewer, but great questions as an interviewer of an expert.
‘Thinking about things just to think about things’. I feel no wiser words have been said. This I think is why I fng love maths
Asking him for an example of a different sized aleph was an excellent question. I love this channel/interviewer.
The animations were so helpful to get a grasp on these ideas
A light saber, Douglas Adams and a Klein bottle: this is a true gentleman.
Two Klein bottles in fact, the coke bottle is one as well!
Well, 3 dimensional analogs of Klein bottles at least
I see the light saber and the (canonical) Klein bottle...where is the Adams reference?
you missed the gameboy
@@seanbirtwistle649 The Ultimate Tetris Machine
I think we can say that there's something special about 2^Aleph_0.
Aleph_0 is the cardinality of the natural numbers N,. "2^N" represents the space of functions from N to a two-element set. W.L.O.G. we can take that set to be the set {TRUE, FALSE} - which helps us to see that each way to map N to {TRUE, FALSE} (i.e. each possible such function) defines a subset of the natural numbers - namely the set for which the function returns TRUE.
The totality of all possible ways to map N to {TRUE, FALSE} then defines all possible subsets of the natural numbers. This means that the cardinality of this function space *is* the cardinality of the powerset (set of all subsets) of N, P(N). Since Aleph_0 is the cardinality of the naturals, the cardinality of their powerset is exactly 2^Aleph_0
As we have learned in the video, this is the cardinality of the reals - so the cardinality of the reals is exactly the cardinality of the powerset of the naturals.
I really appreciate Brady's ability to ask questions that a) I also find fascinating and b) the interviewees really appreciate and can build on. A great skill.
My goodness, I've fallen in love with that mathematician! He was/is so appealing in his intelligence and his amazing ability to describe complex thoughts and theories. Wow. Very appealing man. lol
I think this could become a series, maybe even including several presenters, who talk about the topics of infinities, set theory, axiom of choice, continuum hypothesis... As a physicist, it's the kind of maths I've not been exposed to but would be really interested in learning more about!
This is definitely one of my new favorite Numberphile videos. I really enjoyed the mathematical philosophy talk.
An infinity video about set theory where countability is not once mentioned. Truly an amazing feat. Every time they talked about Aleph not, I expected them to mention that it is a countable infinity.
I think the surreal numbers are really useful for wrapping your head around these different kinds of infinity. IIRC, the sort of sense in which all of the alephs get treated like they're just a different set of natural numbers is a key part of that: you've got all the real numbers on their line, then for each real number there are as many infinitesimal numbers, that are each closer to that real number than any other real number, as there are real numbers; and so on for each of those infinitesimals, etc, all the way down forever; but also in the other direction, all of those original real numbers are closer to a given transfinite number than any other transfinite number is, and there are as many of that class of transfinite numbers as there are reals, each with a whole "real number line" of its own that are closer to that one than to any other; and so on for all of those transfinite numbers, they're all closer to some even greater transfinite number than any other number of that higher class is, etc, all the way up forever.
I think it's provable that the surreal numbers are *the* most complete number line there could possibly be: any kind of number no matter how big or small anyone might ever come up with, it's already in the surreals.
But then what about numbers that aren't on lines? Complex numbers, and hypercomplex numbers like quaternions and octonions. Those can be "sur" as well, not just the reals! There are surcomplex numbers too, and surhypercomplex numbers like surquaternions and suroctonions. I would love to see a video with someone quickly going over the construction from the empty set all the way up to the suroctonions.
It's not true that the surreals contain every number, in part because "number" is not a well-defined term. The complex numbers aren't contained in the surreals, for example. What is contained in the surreals is every ordered field.
@@galoomba5559 Thank you for clarifying that. I did mean to exclude (hyper)complex numbers from the surreals myself, when I said specifically "number *line*", and then went on to talk about "numbers that aren't on lines".
If number line means an ordered field, the surreals are the most complete number line in that for any other (set-sized) number line, you can find a subset of the surreals that is isomorphic to it. The well-ordered transfinite hierarchy of the ordinals (and cardinals) has to come before the surreal numbers (which are non-well-ordered) can be defined though, because the surreals are defined by induction along the ordinals, and the induction requires well-ordering.
In some systems used to study combinatorial game theory, there are numbers (or nimbers) which are larger in magnitude than any surreal, and numbers which are smaller in magnitude than any surreal, and numbers which are not comparable to (some) surreals. You might check out "Winning Ways for your Mathematical Plays," which, like Knuth's book on the surreals, is kind of fun to read even if you don't care too much about the math.
The moment when Brady knew the exact title of the video, and nothing in the world could stop him… priceless.
More of this guy please. He really seems to understand and really knows how to explain his expertise.
This was one of your most impactful videos, i am sure! This is such a gem to think about!
At 14:50 that was such an amazing question, i loved it.
Brady’s questions almost always leave me impressed with both how sharp he is and how good he is at leading the expert on to explain things in a manner more comprehensible to the viewers. He’s sneakily very, very smart.
17:45 - nice job brady. you didn't tell him you already knew.
and that actually is a nice practice, try to think of why those things should feel more natural.
then you take a notice of how things change because of that, and with that you can use other things to identify mistakes or problems with your line of thought.
6:02 What a great question. I can't believe I never thought of it myself, but I'm so glad Brady did. Greatest mathematics journalist of all time, for all history.
The fact that you can have Aleph_0 and Aleph_1, both different magnitudes of Infinity, the set of Alephs is countable, and there is an uncountable Aleph, you've now created a set of Alephs. Absolute Infinity is the Proper Class of these objects which is uncountable. It seems like that is just a concept which nullifies any further expansion of Aleph_Omega Sets because we don't have an abstraction which requires any distinction. There are no properties of that Proper Class which make it unique to another Proper Class of Absolute Infinity, so therefore they are equivalent. What I'm unclear about is why we'd need the distinction between Aleph_Omega and Aleph_Omega+1. Is it just because we've decided that the set of Aleph_Omega is countable because it is defined in terms of Natural numbers and therefor countable? Is the notion of Absolute Infinity thereby an artificial construct of our definition, or is there a necessary reason for us to have this distinction? It seems like we could have simply recognized that Aleph_Omega and Aleph_Omega+1 are members of an uncountable set for which we notate using an set of natural numbers which we define as uncountable. Otherwise what is to prevent Aleph_(Omega+Aleph_(Omega+...))?
My thoughts exactly. A lot of "maths" involving infinity becomes absurd and/or quite subjective, I always think twice about the conclusions they postulate
You need to construct sets from other sets. This is the limitation sets to prevent Russell's paradox. Higher infinities some of them cannot be constructed from below so they are no longer sets but they still have describable properties so they are Classes.
Fantastic. Could listen to these two talk all day.
0:21
Asaf: This is just infinity.
Brady *shocked*: Just infinity!? You say it like its just aa trivial thing.
Asaf *without hesitation*: Yes.
It really is. In some sense it is the second most trivial thing next to nothing.
@@davidwuhrer6704 Nothing is something, because it is nothing. :)
@@c1arkj No, nothing is nothing. The concept of nothing is something.
@@davidwuhrer6704 The fact there is a concept of nothing, makes it something. There is no such thing as nothing. The concept of nothing is nothing.
@@c1arkj You are saying that nothing is something, but the concept of nothing is not. That's saying that 0 = 1, 1 = 0, 1 ≠ 1, 0 ≠ 0.
Do you see the problem with that?
this was a gem of a video. Asaf explaining things clearly (as clearly as he can while keeping it understandable for us!), Brady asking exactly the sort of questions that were needed...
One of the first things you learn in maths is that infinity is not scary, it's just another concept.
Nice pfp
Your 1st grade was wild, man
Hmm. First thing I learned was zero. "There are no more cookies."
Lol, you guys made me realise how funny this comment was. I read it and assumed it meant one of the first things WHEN YOU GET INTO THE MATH COMMUNITY
There are mathematicians who are finitists, who insist that any math done with infinity is not legitimate. And there are even ultrafinitists who insist that very large finite numbers (far bigger than anything that would come up in a physical context) are not "real" or legitimate in some sense.
It's a minority position though.
Astronaut meme:
"Wait, it's all empty sets?"
Always has been.
The empty set is empty, but the set of the empty set is not, it contains the empty set.
Put another way: There is nothing in the empty set, but the set itself is not nothing. (And from that, everything else follows.)
@@davidwuhrer6704 shut up, nerd
16:00 I would love to hear more from Asaf about these even larger infinities and how they are constructed please.
Loved this video! I am actually using Aleph numbers for my epic poetry! I appreciate all your amazing videos.
First time I read about omega, omega+1, etc. until finally epsilon, was in Hofstadters _Gödel, Escher, Bach_ . At the time I wasn't even sure if this wasn't just another wordplay between Achilles and the Tortoise. Glad that's sorted out. 😊
Love Asaf, he's great at making things easy to understand.
Cantor used 2 because in finite sets the set of all subsets has 2 elements, its called power set. It was an abstract generalization.
Studied law and chemistry. Listening to this makes me amaze how people can engage generations of scientists with word plays that lets them explore numbers again as if they were new.
Guys, infinity is just 8 times i
😂
What?
@@chaman9537 multiplying stuff by i is sometimes understood as a 90° rotation. If you rotate 8 by 90° you get ∞
😦😦😦🤯NO WAYYY 8i`=∞!!!!¡
Actually i^3
I love the questions that Brady asks in this video. Exactly the questions I had myself.
I don't think 19:05 is going to be long enough for this man's objective.
I always love how Brady asks the exact questions I'm thinking in my head. I never feel frustrated by a numberphile video, because it's almost like I'm conversing directly with the mathematician. It's uncanny how good he is at asking the questions we're all thinking.
We see that original Gameboy in the background.
Behind the light saber
always love seeing my namesake as the topic of a video.
Was it intentional? 🦭
@@AlanShi-wm6tw my username? yeah, a friend lent me a book about Georg Cantor years and years ago, and i became fascinated by his work and the concept of there being more than one infinity, and eventually just started using either alephnull or alefnull as my username. even got the aleph null symbol tattooed on my wrist lol
More asaf videos on logic and discrete math please!
So rare I actually feel a little smarter after watching this video.. This is the very best most understandable declarative explanation of set theory I have ever seen because you can almost touch the empirical with this explanation.
About the "why two?" question around 3:46, there's actually a pretty natural intuition for it.
If you try to create a subset out of a set A, you can do so with the application f: x -> { 0 if x not in subset, 1 otherwise
That is, one of two possibilities for each item.
That is, 2^|A| possible subsets.
So the powerset of A has 2^|A| element in it, and for N to R numbers we can prove the relation between their size is that of getting powerset, so it "makes sense" to use the notation.
Not sure how well this continues holding for higher infinities but I would expect it to sortof continue making sense. 3^ might also work but wouldn't be minimal.
4:17 They use 2 because of sets. It's the same as the power set just with an exact equivalence. There are two options: in the set vs. not in the set.
Exponentiation in set theory: x^y is the set of functions from y to x. This is why in set theory and combinatorics 0^0=1
The infinite and the infinitesimal always fascinates me. The symbol for ♾️ is an interesting subject.
I love mathematics, infinity--and absolute infinity!!! :) :) :)
I was always fascinated by the fact that there are more real numbers between 0 and 1 then there are counting numbers, which is like the simplest example for differently sized infinities
Math is like a boys club: size matters
Thanks for all the help during undergrad, Asaf
מזהה את המבטא ומתלהבת
בודקת קורות חיים
אשמח להתייעצות כסטודנטית מתחילה למתמטיקה 😂
I usually only understand about half of the concepts, but i feel more knowledgeable for the fact of watching these videos.
I am requesting a video on ALGEBRAIC TOPOLOGY or DIFFERENTIAL GEOMETRY. I am waiting……..
9:59 I love that answer.
I would also recommend Vsauce's video, 'counting past infinity' additional information.
Second!
Saw a Vsauce video on it long time ago. Glad to see Numberphile covering it now!
Very questionable how this can be relevant to reality. Can anyone point me to how this is useful?
It isn't. Still interesting despite it's lack of purpose, like most things in life.
That was a big mind blower moment in undergraduate mathematics for me. First they showed us Cantor's diagonal argument to prove the Rational numbers were strictly smaller than the Reals. Then they proved that between any two real numbers there is a rational number. Completely counterintuitive and shows how intuition breaks down significantly when talking about infinity.
For some reason this video gave me anxiety. Am I the only one? I better stop watching now.
Math is great😅
I have caught a glimpse of the shadow of infinity three times in my life, thus far. It is an overwhelming experience which one does not quickly forget.
@@curtiswfranks Maybe it's better I don't ask what that was about.
My brain started hurting when he said that 2 to the power of aleph nought was the same as 3 to the power of aleph nought, and that there are as many Real numbers as there are Complex numbers.
When I slice the sphere of Netscape Navigator, I'm only limited by my imagination as to how many "types" of shapes I see: whether it is a ceramic pot, a dumbbells or an igloo.
Study of infinite sets is the string theory of mathematics. Not even wrong. The "answer" starting at 13:30 is just word salad
"You don't understand things. You just get used to them." Sigh. I have trouble getting used to THAT!
Zero = -
fish =
no
\{°◇°}/ -[~_~]-
Brady, thanks for always asking the question I am thinking.
First!
how!? it came out 15 seconds ago >:(
Patrons got an early notification 😂 but I always wonder that myself when I see three comments on a video that just released @@aguyontheinternet8436
The towers of hanoi analog was quite crucial to comprehending the whole thing.
Those symbols and names I will never be able to intuitively remember or name at a later date, nor will I comprehend everything.
Free Palestine
Free plastilin. GTFO!
From whom? The terrorists hiding amongst the civilians?
There is no place called palestine... it's just a fake slogan
@@eem19584 lol, keep telling yourself that
"There is no war in ba sign se"@@eem19584
Best math channel in the universe!
“Why’d you use 2; because it’s the 1st number?” That could be its own episode.
Just watched the infinity episode from 5 yr ago. It seems if you ask 10 different people what infinity is, you'll get 10 different interpretations
What an incredible video. Asaf, Brady, thank you. thank you.
Man, all this stuff about elephants and infinity goes right over my head
I liked the way this ended. I've thought for a long time that part of the reason analysis is at first so difficult and frustrating is because it's almost impossible to wrap one's head around the fact that the natural numbers are dense in the reals but come from an infinitely smaller set.
The animation for the reordering is absolutely great!
That was a beautiful talk. Thank you so much!
Just let this man speak and chop it in a 100 videos series, and I will watch them all.